Escaping the Krylov space during the finite-precision Lanczos algorithm

The Lanczos algorithm, introduced by Cornelius Lanczos, has been known for a long time and is widely used in computational physics. While often employed to approximate extreme eigenvalues and eigenvectors of an operator, interest in the sequence of basis vectors produced by the algorithm has been recently increased in the context of Krylov complexity. Although it is generally accepted and partially proven that the procedure is numerically stable for approximating the eigenvalues, there are numerical problems when investigating the Krylov basis constructed via the Lanczos procedure. In this paper, we show that the loss of orthogonality and the attempt of reorthogonalization fall short of understanding and addressing the problem. Instead, the sequence of numerical Lanczos vectors in finite-precision arithmetic escapes the true vector space spanned by the exact Lanczos vectors. This poses a real threat to the interpretation in view of the operator growth hypothesis.